A dynamic study of the high-speed oblique water entry of a stepped cylindrical-cone projectile


High-speed oblique water entry is an interesting subject, many physical aspects of which remain unknown up to now. Among high-speed air-to-water projectiles, the supercavitating cylindrical-cone (SCC) ones have economic and operational advantages over the other types. However, maintaining stability of the SCC projectiles inside the cavity at shallow entry angles is a challenging issue from both practical and design-related points. The first section of the present study proposes a novel and unique scheme of air-to-water supercavitating projectile design which is called the supercavitating stepped cylindrical-cone (SSCC) projectile. The SSCC scheme is analyzed numerically to investigate the projectile stability improvement at shallow entry angles. The 6DOF dynamics of the SSCC projectile are investigated using the Star-CCM+ commercial code in the presence of three phases of air, water and vapor in a three-dimensional and transient model. Accuracy of numerical results and the model’s ability to simulate the physical phenomena of water entry is validated using experimental results from the literature, and both are in good agreement. In the present study, the high-speed oblique water entry dynamics of the SSCC projectile are investigated for five certain entry angles varying from 10° to 60°. The results show that the SSCC projectile faces intensive unstabilizing forces in the water entry process which leads to a heavy pitching moment and, hence, intensive angular velocity (\( \dot{\gamma } \)) on the projectile. This study also proves that the presence of step enhances the projectile stability in the entry process. The present study shows that, based on their geometry and mass characteristics, each SSCC projectile is capable of withstanding instability up to a critical value of the angular velocity (\( \dot{\gamma }_{\text{Cr}} \)). Therefore, projectile stability inside the cavity can be achieved when the value of maximum angular velocity (\( \left| {\dot{\gamma }} \right|_{\hbox{max} } \)) experienced by the projectile is lower that \( \dot{\gamma }_{\text{Cr}} \) (i.e., \( \left| {\dot{\gamma }} \right|_{\hbox{max} } < \dot{\gamma }_{\text{Cr}} \)). The results of this study also show that \( \left| {\dot{\gamma }} \right|_{\hbox{max} } \) is inversely correlated with the \( \gamma \), and that it follows a simple equation which is proposed in this study. Therefore, projectile stability inside the cavity can also be practically achieved by adjusting the shooting mechanism at an angle higher than the minimum stable entry angle (\( \gamma_{\hbox{min} } \)). This study also proposes an effective numerical approach to evaluate \( \gamma_{\hbox{min} } \) of a supercavitating projectile. It should be noted that determining the value of \( \gamma_{\hbox{min} } \) is an important factor from both a practical and design-related points of view.

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F :

External force vector exerted to the projectile

fn b :

Force applied to the projectile surface number n, along vector b (Fig. 5)

g :

Earth’s gravitational acceleration vector

I :

Moment of inertia

m n :

Pitching moment applied to the projectile surface number n, along z (Fig. 5)

m p :

Projectile mass

\( \dot{m}_{\text{qk}} \) :

Mass transfer rate from phase q to phase k

M :

Hydrodynamic moment vector exerted to the projectile

p :

Static pressure

\( p_{\text{sat}} \) :

Saturation pressure

\( p_{\infty } \) :

Ambient pressure

R :

Cavity radius

V :

Projectile velocity vector

\( \left( {V_{X} ,V_{Y} , V_{Z} } \right) \) :

Projectile velocity component along the inertial coordinate system axes

\( \left( {V_{x} ,V_{y} , V_{z} } \right) \) :

Projectile velocity component along the body coordinate system axes

\( \left( {X,Y,X} \right) \) :

Position of the center of mass of the projectile along the inertial coordinate system axes (Fig. 4)

\( \left( {x,y,z} \right) \) :

Position of the center of mass of the projectile along the body coordinate system axes (Fig. 4)

(\( \gamma ,\varphi ,\psi \)):

The rotational angles of projectile with respect to the coordinate axis

\( \dot{\gamma } \) :

Projectile angular velocity

θ :

Projectile velocity angle with respect to the water surface

\( \alpha \) :

Angle of attack

\( \omega \) :

Angular velocity

\( \alpha_{\text{l}} ,\alpha_{\text{v}} ,\alpha_{\text{a}} \) :

Volume fraction of liquid, vapor and air phases

\( \rho_{\text{l}} ,\rho_{\text{v}} ,\rho_{\text{a}} \) :

Density of liquid, vapor and air phases

\( \mu \) :

Dynamic viscosity

\( \rho_{\text{w}} \) :

Water density


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Akbari, M.A., Mohammadi, J. & Fereidooni, J. A dynamic study of the high-speed oblique water entry of a stepped cylindrical-cone projectile. J Braz. Soc. Mech. Sci. Eng. 43, 2 (2021). https://doi.org/10.1007/s40430-020-02727-2

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  • Oblique water entry
  • SSCC projectile
  • Supercavitation
  • Stability